Signless Laplacian determinations of some graphs with independent edges

{Signless Laplacian determinations of some graphs with independent edges}% {Let $G$ be a simple undirected graph. Then the signless Laplacian matrix of $G$ is defined as $D_G + A_G$ in which $D_G$ and $A_G$ denote the degree matrix and the adjacency matrix of $G$, respectively. The graph $G$ is said to be determined by its signless Laplacian spectrum ({\rm DQS}, for short), if any graph having the same signless Laplacian spectrum as $G$ is isomorphic to $G$. We show that $G\sqcup rK_2$ is determined by its signless Laplacian spectra under certain conditions, where $r$ and $K_2$ denote a natural number and the complete graph on two vertices, respectively. Applying these results, some {\rm DQS} graphs with independent edges are obtained

Laplacian Spectral Characterization of Some Unicyclic Graphs

Let W(n;q,m1,m2) be the unicyclic graph with n vertices obtained by attaching two paths of lengths m1 and m2 at two adjacent vertices of cycle Cq. Let U(n;q,m1,m2,…,ms) be the unicyclic graph with n vertices obtained by attaching s paths of lengths m1,m2,…,ms at the same vertex of cycle Cq. In this paper, we prove that W(n;q,m1,m2) and U(n;q,m1,m2,…,ms) are determined by their Laplacian spectra when q is even

On the uniqueness of the Laplacian spectra of coalescence of complete graphs

Acknowledgments: I would like to thank Nicole Snashall for her encouragement, comments and support. I also would like to thank Jozef Siran for his comments on amalgamations, which led to more thinking about defining my graphs. This project started with an application to social networks; however, it developed into a much more general study. I would like to dedicate this paper to the memory of Alto Zeitler (1945-2022), my mathematics teacher.Peer reviewedPublisher PD

Spectral characterizations of signed lollipop graphs

Let Γ=(G,σ) be a signed graph, where G is the underlying simple graph and σ:E(G)→{+,-} is the sign function on the edges of G. In this paper we consider the spectral characterization problem extended to the adjacency matrix and Laplacian matrix of signed graphs. After giving some basic results, we study the spectral determination of signed lollipop graphs, and we show that any signed lollipop graph is determined by the spectrum of its Laplacian matrix

Signless Laplacian determinations of some graphs with independent edges

Let $G$ be a simple undirected graph. Then the signless Laplacian matrix of $G$ is defined as $D_G + A_G$ in which $D_G$ and $A_G$ denote the degree matrix and the adjacency matrix of $G$, respectively. The graph $G$ is said to be determined by its signless Laplacian spectrum (DQS, for short), if any graph having the same signless Laplacian spectrum as $G$ is isomorphic to $G$. We show that $G\sqcup rK_2$ is determined by its signless Laplacian spectra under certain conditions, where $r$ and $K_2$ denote a natural number and the complete graph on two vertices, respectively. Applying these results, some DQS graphs with independent edges are obtained